Development of focus points in the wing disc during eyespot determination.

<p>(a) Time series of <i>Notch</i> expression patterns in <i>Junonia coenia</i> wing discs for the final instar eyespot determination. The <i>Notch</i> expression patterns were obtained by anti-<i>N</i> mouse monoclonal antibody and were visualized on a fluorescent light microscope [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141434#pone.0141434.ref018" target="_blank">18</a>]. (Upper row) The five panels show stained wing discs. (Bottom row) The five panels show the wing cells extracted from the respective figures in the upper panels. Regarding the orientation of bottom panels, the upper side corresponds to the proximal boundary and the bottom side corresponds to the distal boundary of the wing cell, respectively. Insets in the panels detail gene expression in the wing cells marked by white arrows. (b) The corresponding adult forewing of <i>J</i>.<i>coenia</i>. (c) Simulation results of Fig 7 (a) by use of Eq (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141434#pone.0141434.e003" target="_blank">3.1</a>). The initial data and boundary conditions are perturbed by uniformly distributed noise which leaves the qualitative features of the results unchanged. In Fig 7 (a), we could see a migration of the focal point into the distal direction from the 3^rd stage (middle) to the 4^th stage (next to the middle). Both photos (a) and (b): courtesy of Dr. Robert Reed of Cornell University.</p

Proximal boundary conditions may govern eyespot focus point determination.

<p>The figure shows snapshots of the activator concentration corresponding to the solution of Eq (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141434#pone.0141434.e003" target="_blank">3.1</a>). The boundary conditions on the proximal boundary (top) of the rectangular cell for the activator are of the form <i>k</i>_<i>p</i><i>a</i>₁^<i>ss</i> where <i>k</i>_<i>p</i> = 0, 1 and 2 (reading from left to right in each row) and <i>a</i>₁^<i>ss</i> is the (activator) steady state value. The veins (left and right boundaries of each wing cell) have Dirichlet (fixed) boundary conditions for the activator with constant values at twice the steady state. Initially in all the wing cells a vertical stripe of high activator concentration is generated originating from the zero-flux distal boundary (bottom). In the wing cells with lowest activator values at the proximal boundary (left hand), a spot forms and this spot eventually moves towards the center of the cell (see also Section <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141434#sec013" target="_blank">3.3</a>). In the wing cells with medium activator values at the proximal boundary (middle), we have both the formation of a spot from the receding midline peak and later the insertion of a new spot that originates from the proximal boundary with the steady state consisting of two spots. In the wing cell with highest activator values at the proximal boundary (right hand), the vertical stripe recedes without leaving behind a spot.</p

Matlab_code

Matlab codes used to generate all the numerical and statistical analysis carried out in the manuscript

Incomplete vein development leaves two focus points with an eyespot covering two focus points.

<p>(a) Normal (left) and abnormal (right) eyespot patterns on the hind wing of the butterfly <i>Ypthima arugus</i>. (b) Sketch of the abnormal venation system and an arrow to show two distinct focus points. (c) Simulations of the abnormal case of incomplete vein development shown in (a) (right) by use of Eq (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141434#pone.0141434.e003" target="_blank">3.1</a>). This incomplete vein development leads to two focus points forming close to both the incompletely developed vein’s end point. The eventual pattern observed on the butterfly wing is that of a single eyespot generated by two focus points that are in close proximity. The corresponding normal pattern is of two distinct eyespots with orally separated foci. Photos (a) and the sketch (b): courtesy of Mr.Toru Tokiwa.</p

Parameter values used for all the simulations of Eq (3.1).

<p>Parameter values used for all the simulations of Eq (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141434#pone.0141434.e003" target="_blank">3.1</a>).</p

Steady state values of the activator concentration in simulations of Eq (3.1) on a domain of increasing width in the proximal-distal direction (top to bottom) and with curved proximal (top) and wing margin (bottom) boundaries.

<p>The left hand figure corresponds to constant boundary conditions equal to zero on the proximal boundary curve. The right hand figure corresponds to proximal boundary conditions equal to twice the activator steady state. The observed behavior is analogous to the rectangular domain case.</p

Ventral eyespot patterns of the butterfly <i>Ypthima arugus</i> (Nymphalidae, Satyrinae) at rest (left), and the extended adult specimen (right).

<p>The right-hand side photo: courtesy of Mr.Toru Tokiwa.</p

Development of eyespot focus points in the wing disc of <i>Junonia coenia</i> (Nymphalidae, Nymphalinae).

<p>Numbers 1~7 in the photos show a time course of the <i>Notch</i> expression pattern during the focus point development. The expression pattern by antibody staining were visualized on a fluorescent light microscope and digitally photographed. Black arrows in photo numbers 1, 2, and 5 indicate pre-veins, which finally evolve to become veins of the adult butterfly wing. White arrows in photo 6 show two peaks of <i>N</i>-related chemicals along the centerline of each wing cell, the right-hand one of which evolves into a focus point afterwards (in photo number 7) while no focus point remains on the left-hand wing cell. Photos: courtesy of Prof. Fred Nijhout of Duke University. For more details on the adult forewing of <i>J</i>.<i>coenia</i> butterfly, see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141434#pone.0141434.g007" target="_blank">Fig 7</a> in Section <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141434#sec013" target="_blank">3.3</a>.</p

Focus points on the dorsal hind wing of <i>Precis coenia</i> and numerical simulation results by the 2-stage model.

<p>(a) Steady state values of activator concentration (<i>u</i>₁) of the 1<i>D</i> RDS (4.1) with a constant value of the function <i>γ</i>(<i>x</i>). (b) Dosal hindwing of <i>P</i>. <i>coenia</i>. (c) Steady state values of the activator concentration (<i>a</i>₁) for the seven independent bulk RDSs (3.1) with proximal boundary conditions given by (1/3)<i>u</i>₁ where <i>u</i>₁ is the steady state activator concentration shown above. (d) The hind wing imaginal disc of <i>P</i>.<i>coenia</i> with focus points labelled. Two white arrows point two <i>Dll</i> stained focus points. (Left hand column) Simulation results of the 2-stage model for focus point formation with aconstant value of the reaction rate <i>γ</i> appearing in Eq (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141434#pone.0141434.e018" target="_blank">4.1</a>) (<i>γ</i>(<i>x</i>) = 5.4). The model generates the formation of foci in wing cells 2 and 5 and no foci in the other wing cells similar to the experimental observations. (Right hand column) The adult <i>P</i>. <i>coenia</i> dorsal hindwing with two eyespots (top) and the fifth-instar hindwing imaginal disc displaying a pre-pattern with two foci (bottom), which correspond to eyespots positions on the adult dorsal hindwing. Experimental figures: from Brakefield et al. [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141434#pone.0141434.ref002" target="_blank">2</a>] with permission by the publisher.</p

Focus points on the ventral hind wing of <i>Bycyclus anynana</i> and numerical simulation results by the 2-stage model.

<p>(a) Steady state values of activator concentration (<i>u</i>₁) of the 1<i>D</i> RDS (4.1) with <i>γ</i>(<i>x</i>) = 0.01. (b) Ventral hindwing of <i>B</i>. <i>anynana</i>. (c) Steady state values of the activator concentration (<i>a</i>₁) for the seven independent bulk RDSs (3.1) with proximal boundary conditions given by (1/3)<i>u</i>₁ where <i>u</i>₁ is the steady state activator concentration shown above. (d) The hind wing imaginal disc of <i>B</i>. <i>anynana</i> with focus points labelled. (Left hand column) Simulation results of the 2-stage model for focus point formation with a small constant value of the reaction rate <i>γ</i> appearing in the 1<i>D</i> RDS (4.1) (<i>γ</i>(<i>x</i>) = 0.01). The model generates a focus point in every wing cell. (Right hand column) The adult ventral hind wing of <i>B</i>. <i>anynana</i> with seven eyes-pots (top) and the fifth-instar hind wing imaginal disc displaying a pre-pattern with seven foci (bottom), which correspond to eyespots positions on the adult ventral hind wing. Experimental figures: from Brakefield et al. [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0141434#pone.0141434.ref002" target="_blank">2</a>] with permission by the publisher.</p